Hardy Type Asymptotics for Cosine Series in Several Variables with Decreasing Power-Like Coefficients

摘要

The investigation of the asymptotic behavior of trigonometric series near theorigin is a prominent topic in mathematical analysis. For trigonometric seriesin one variable, this problem was exhaustively studied by various authors in aseries of publications dating back to the work of G. H. Hardy, 1928. Trigonometric series in several variables have got less attention. The aim ofthe work is to partially fill this gap by finding the asymptotics oftrigonometric series in several variables with the terms, having a form of `oneminus the cosine' up to a decreasing power-like factor: [sum_{zinmathbb{Z}^{d}setminus{0}}rac{1}{|z|^{d+lpha}}left(1-coslanglez,hetaangleight), qquad hetainmathbb{R}^{d}, ] where$langlecdot,cdotangle$ is the standard inner product and $|cdot|$ isthe max-norm on $mathbb{R}^{d}$. The approach developed in the paper is quite elementary and essentiallyalgebraic. It does not rely on the classic machinery of the asymptotic analysissuch as slowly varying functions, Tauberian theorems or the Abel transform.However, in our case, it allows to obtain explicit expressions for theasymptotics and to extend to the general case $dge 1$ classical results of G.H. Hardy and other authors known for $d=1$.